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Persistence Modules on Commutative Ladders of Finite Type

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Abstract

We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander–Reiten theory is applied to develop the theoretical and algorithmic foundations. In particular, we prove that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander–Reiten quivers. Furthermore, a generalization of persistence diagrams is introduced by using Auslander–Reiten quivers. We provide an algorithm for computing persistence diagrams for the commutative ladders of length 3 by using the structure of Auslander–Reiten quivers.

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Acknowledgments

The authors would like to thank Hideto Asashiba, Hiroyuki Ochiai, and Dai Tamaki for valuable discussions and comments. This work is partially supported by JSPS Grant-in-Aid (24684007) and JST Mathematics CREST (15656429).

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Emerson G. Escolar

  2. WPI - Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Yasuaki Hiraoka

Authors
  1. Emerson G. Escolar
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  2. Yasuaki Hiraoka
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Correspondence to Yasuaki Hiraoka.

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Editor in Charge: Herbert Edelsbrunner

Appendix: The Auslander–Reiten Quivers of \(\textit{CL}(\tau _n)\) with \(n\le 4\)

Appendix: The Auslander–Reiten Quivers of \(\textit{CL}(\tau _n)\) with \(n\le 4\)

See Figs. 13, 14, 15, 16, 17, 18, 19, and 20.

Fig. 13
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The Auslander–Reiten quiver of \(\textit{CL}(f)\)

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Fig. 14
figure 14

The Auslander–Reiten quiver of \(\textit{CL}(ff)\)

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The Auslander–Reiten quiver of \(\textit{CL}(fb)\)

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The Auslander–Reiten quiver of \(\textit{CL}(bf)\)

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Fig. 17
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The Auslander–Reiten quiver of \(\textit{CL}(fff)\)

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Fig. 18
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The Auslander–Reiten quiver of \(\textit{CL}(fbb)\)

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Fig. 19
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The Auslander–Reiten quiver of \(\textit{CL}(bfb)\)

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Fig. 20
figure 20

The Auslander–Reiten quiver of \(\textit{CL}(bbf)\)

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Escolar, E.G., Hiraoka, Y. Persistence Modules on Commutative Ladders of Finite Type. Discrete Comput Geom 55, 100–157 (2016). https://doi.org/10.1007/s00454-015-9746-2

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